ORIGINAL PAPER

Higher order amplitude squeezing in fourth and fifth harmonicgeneration

S Gill1*, S Rani2 and N Singh3

1Department of Applied Physics, University Institute of Engineering and Technology, Kurukshetra 136 119, Haryana, India

2Department of Applied Physics, S K Institute of Engineering and Technology, Kurukshetra 136 118, Haryana, India

3Department of Physics, Kurukshetra University, Kurukshetra 136 119, Haryana, India

Received: 28 January 2011 / Accepted: 19 August 2011 / Published online: 5 June 2012

Abstract: The quantum effect of squeezing of electromagnetic field has been investigated in the fifth order amplitude in

fundamental mode in fourth and fifth harmonic generation under the short time approximation. It has been observed that

squeezing depends on coupling constant g and phase of the field amplitude. The effect of photon number on squeezing and

the variation of signal-to-noise ratio in fifth order of field amplitude for a squeezed state with photon number has also been

investigated.

Keywords: Harmonic generation; Squeezing; Photon statistics

PACS No.: 42.65.-k

1. Introduction

Squeezed states of electromagnetic field are non-classical

in nature and constitute an interesting subject of quantum

optics. Squeezed states of field are special class of mini-

mum uncertainty states with reduced noise in one of the

quadrature component at the expense of increased noise in

the complementary quadrature component as compared to

that of coherent states. In a coherent state, the fluctuations

in the quadrature components are equal and uncertainty in

their product is the minimum given by Heisenberg’s

uncertainty relation. The quantum fluctuations in a coher-

ent state are equal to zero point fluctuations and are ran-

domly distributed in field quadrature components but

squeezed states no longer have their quantum noise ran-

domly distributed in the quadrature components.

Initial studies regarding squeezing have been done

keeping in view the theoretical aspects of the idea [1, 2].

However, only after applications for squeezed light in 1980s

squeezing has been discussed in more detail. Squeezed states

are observed in many non linear optical processes such as

harmonic generation [3, 4], multi wave mixing [5, 6], non

linear polarization rotation [7, 8], optical parametric oscil-

lation [9–11], Raman and hyper Raman processes [12–14].

The quantum nature of the electromagnetic field imposes

a fundamental limit on the sensitivity of optical precision

measurements. In the world of high precision measurement

squeezed states have several possible potential applications

such as in interferometry for the detection of gravitational

waves [15, 16] as well as in the field of quantum informa-

tion, for example, for quantum teleportation [17–19],

quantum communication [20] and quantum computation

[21]. Additionally there is a renewed interest in squeezed

light with regards to application in the field of continuous

variable quantum networking [22, 23].

The aim of the present work is to study the quantum

squeezing in fourth and fifth harmonic generation in fifth

order of the field amplitude.

2. Squeezing of fifth order amplitude in fundamental

mode in fourth harmonic generation

Fourth harmonic generation model has been adopted from

the works of Chen et al. [24] and is shown in Fig. 1. In this

model, the interaction is looked upon as a process which

involves absorption of four photons, each having frequency

� 2012 IACS

*Corresponding author, E-mail: savitamaggie@gmail.com

Indian J Phys (May 2012) 86(5):371–375

DOI 10.1007/s12648-012-0060-z

x1 going from state 1j i to state 2j i and emission of one

photon of frequency x2, where x2 ¼ 4x1.

The Hamiltonian for this process is given as follows

ð�h ¼ 1Þ

H ¼ x1aya þ x2byb þ g a4by þ ay4b� �

ð1Þ

in which g is a coupling constant for fourth harmonic

generation. A = a Exp ix1tð Þ and B = b Exp ix2tð Þ are the

slowly varying operators at frequencies x1 and x2, a ay� �

and b by� �

are the usual annihilation (creation) operators,

respectively.

The Heisenberg equation of motion for fundamental

mode A is given as (�h ¼ 1)

dA

dt¼ oA

otþ i H;A½ � ð2Þ

Using Eqs. (1) in (2), we obtain

A:¼ �4igAy3B ð3Þ

Similarly,

B:¼ �igA4 ð4Þ

By assuming the short time interaction of waves with the

medium and expanding A(t) by using Taylors series

expansion and retaining the terms up to g2t2 as

A tð Þ ¼ A � 4igtAy3B þ 2g2t2

� 12Ay2A3 þ 36AyA2 þ 24A� �

ByB � Ay3A4h i

ð5Þ

Fifth order amplitude of fundament mode is expressed as

A5 tð Þ ¼ A5 � 20igt

� Ay3A4 þ 6Ay2A3 þ 12AyA2 þ 6A� �

B

� 10g2t2 Ay3A8 þ 6Ay2A7�

þ 12AyA6 þ 6A5�

ð6Þ

Initially, we have considered the quantum state of field

amplitude as a product of coherent state for the fundamental

mode A and the vacuum state for the harmonic mode B i.e.

wj i ¼ aj i 0j i ð7Þ

Using Hillery’s approach [2] for higher order squeezing,

the real quadrature component of fifth order amplitude in

fundamental mode A is given as

Z1A tð Þ ¼ 1

2A5 tð Þ þ Ay5 tð Þh i

¼ 1

2

�A5 þ Ay5 � 10g2t2

�Ay3A8 þ 6Ay2A7

þ 12AyA6 þ 6A5 þ Ay8A3 þ 6Ay7A2

þ 12Ay6A þ 6Ay5��

ð8Þ

Using Eqs. (7) and (8), we get

DZ1A tð Þ½ �2¼ wh jZ21A tð Þ wj i � wh jZ1A tð Þ wj i2

¼ 1

225½ aj j8þ200 aj j6þ600 aj j4þ600 aj j2

þ 120� 20g2t2ð7:5 aj j4a10 þ 60 aj j2a10

þ 120a10 þ 7:5 aj j4a�10 þ 60 aj j2a�10

þ 120a�10 þ 20 aj j14þ210 aj j12

þ 720 aj j10þ720 aj j8Þ�

ð9Þ

where a is a dimensionless complex number given by

a ¼ aj jExpðihÞUsing Eqs. (5) and (7), number of photons in mode

A may be expressed as

NA tð Þ ¼Ay tð ÞA tð Þ

¼AyA � 4igt Ay4B � A4By� �

� 4g2t2Ay4A4

ð10Þ

Using Eqs. (7) and (10), we get

1

425N4

A þ 50N3A þ 275N2

A þ 250NA þ 120�

¼ 1

425½ aj j8þ 200 aj j6þ 600 aj j4þ 600 aj j2þ 120

� 20g2t2ð20 aj j14þ 210 aj j12þ 720 aj j10þ 750 aj j8Þ�ð11Þ

Condition for fifth order amplitude squeezing is given

as [2]

ðDZiÞ2 \1

4ð25N4

A þ 50N3A þ 275N2

A þ 250NA þ 120Þ�

for i ¼ 1 or 2

Subtracting Eq. (11) from Eq. (9), we obtain

Fig. 1 Fourth harmonic generation model

372 S. Gill et al.

DZ1A tð Þ½ �2� 1

425N4

A þ 50N3A þ 275N2

A þ 250NA þ 120�

¼ �10g2t2ð7:5 aj j14þ60 aj j12þ120 aj j10Þ cos 10h ð12Þ

where h is the phase angle, with a ¼ aj jExpðihÞ and a� ¼aj jExpð�ihÞ. The right hand side of Eq. (12) is nega-

tive, indicating that squeezing will occur in fifth

order amplitude in fundamental mode for which cos

10h [ 0 :

3. Squeezing of fifth order amplitude in fundamental

mode in fifth harmonic generation

Similar to Sect. 2, we can obtain the following results for

squeezing of fifth order amplitude in fifth harmonic gen-

eration following the works of Chang et al. [25] and is

shown in Fig. 2.

The Hamiltonian for this process is given as follows

(�h ¼ 1)

H ¼ x1aya þ x2byb þ g a5by þ ay5b� �

ð13Þ

Short-time approximated second-order solution of this

Hamiltonian is

A tð Þ ¼ A � 5igtAy4B þ 5

2g2t2

�h5: 4Ay3A4 þ 24Ay2A3 þ 48AyA2 þ 24A� �

ByB

� Ay4A5i

ð14Þ

Using Eqs. (14) and (7) for squeezing of fifth order

amplitude in fundamental A, the real quadrature compo-

nent is

Z1A tð Þ ¼ 1

2

A5 þ Ay5 � 5

2g2t2

�5Ay4A9

þ 40Ay3A8 þ 120Ay2A7 þ 120AyA6 þ 24A5

þ 5Ay9A4 þ 40Ay8A3 þ 120Ay7A2

þ 120Ay6A þ 24Ay5��

ð15Þ

Using Eqs. (14), (15) and (7) a straightforward but

strenuous calculation yields

DZ1A tð Þ½ �2� 1

425N4

A þ 50N3A þ 275N2

A þ 250NA þ 120�

¼ � 125g2t2ð aj j16þ12 aj j14þ48 aj j12þ 60 aj j10Þ cos 10h

ð16Þ

The right hand side of Eq. (16) is negative, indicating

that squeezing will occur in fifth order amplitude in

fundamental mode for which cos 10h [ 0:

4. Signal-to-noise ratio

Signal-to-noise ratio is defined as ratio of the magnitude of

the signal to the magnitude of the noise. With the

approximations h ¼ 0 and gtj j2� 1, the maximum signal-

to-noise ratio (in decibels) in fifth order amplitude in fourth

and fifth harmonic generation, is given below.

Using Eqs. (8) and (9), signal-to-noise ratio in fourth

harmonic generation is defined as

SNR4 ¼ 20� log10

\Z1AðtÞ[ 2

½DZ1AðtÞ�2

SNR4 ¼ 20� log10

ð2 aj j8þ 12 aj j6þ 24 aj j4þ 12 aj j2Þð17:5 aj j6þ 165 aj j4þ 480 aj j2þ 375Þ

Using Eq. (15), Signal-to-noise ratio in fifth harmonic

generation is defined as

SNR5 ¼ 20� log10

ð5 aj j8þ 40 aj j6þ 120 aj j4þ 120 aj j2þ 24Þð10 aj j6þ 120 aj j4þ 480 aj j2þ 600Þ

5. Results

We denote right hand side of Eqs. (12) and (16) by SF and ST

respectively, which are negative within in the domain of the

validity of the solution and thus show the presence of

squeezing in fifth order amplitude in fundamental mode in

fourth and fifth harmonic generation respectively. Taking

gtj j2¼ 10�4 and h ¼ 0 for maximum squeezing, the vari-

ations of SF and ST are shown in Figs. 3 and 4 respectively.

Fig. 2 Fifth harmonic generation model

High squeezing 4th & 5th harmonic generation 373

It is clear from Figs. 3 and 4 that the squeezing increases

non-linearly with aj j2. This confirms that the squeezed

states are associated with the photon number in

fundamental mode. The variation of SNR in fifth order of

field amplitude for a squeezed state with photon number

has also been shown in Fig. 5. The signal-to-noise ratio is

higher in fifth order amplitude squeezed states of fifth

harmonic generation as compared to fourth harmonic

generation.

6. Conclusions

It is shown that the selective phase values of field ampli-

tude of fundamental mode during fourth and fifth harmonic

generation lead to squeezing up to fifth order amplitude.

Further, Figs. 3, 4 and 5 show that the degree of squeezing

as well as signal-to-noise ratio is higher in fifth harmonic

generation as compared to fourth harmonic generation

which can be used in high precision quantum measurement

and in obtaining noise reduction in non linear optical

devices such as interferometers for the detection of gravi-

tational waves and homodyne detectors.

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Fig. 3 Dependence of fifth order amplitude squeezing on aj j2 in

fourth harmonic generation

Fig. 4 Dependence of fifth order amplitude squeezing on aj j2 in fifth

harmonic generation

Fig. 5 Signal-to-noise ratio for fifth order squeezing in fourth and

fifth harmonic generation

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High squeezing 4th & 5th harmonic generation 375